Independent solution

How to solve this Linear Combinations question

Setup

Setup

Let S be the sum of all 49 independent normal amounts. A sum of independent normal variables is normal.

S=∑i=149Xi,S∼N(μS,σS2)S=\sum_{i=1}^{49}X_i,\qquad S\sim N(\mu_S,\sigma_S^2)
μS=122,500\mu_S=122{,}500

Model

Model

Add the independent component variances; the differing component means do not affect the aggregate variance.

σS2=∑i=149Var⁡(Xi)=49(7002)=24,010,000\sigma_S^2=\sum_{i=1}^{49}\operatorname{Var}(X_i)=49(700^2)=24{,}010{,}000
σS=24,010,000=4,900\sigma_S=\sqrt{24{,}010{,}000}=4{,}900

Compute

Compute

Use the 90th-percentile standard-normal value and convert the result to thousands.

z0.90=1.281551566z_{0.90}=1.281551566
q0.90=122,500+1.281551566(4,900)=128,779.6027q_{0.90}=122{,}500+1.281551566(4{,}900)=128{,}779.6027
q0.90=128.7796027 thousandq_{0.90}=128.7796027\text{ thousand}

Answer

Answer

The requested percentile rounds to 129 thousand, which is choice D.

q0.90≈129 thousand(D)\boxed{q_{0.90}\approx 129\text{ thousand}\quad\text{(D)}}